Accurate predictions of annular frictional pressure loss are important for optimal
well bore hydraulic program design. Inaccurate prediction of frictional pressure drop in
the annulus can result in an underestimation of the bottom hole pressure, which might
then exceed the strength of the formation, thus causing loss of drilling fluid and creating a
potentially dangerous situation due to the resulting loss of hydrostatic head.
In this study fully developed laminar axial flow of non-Newtonian fluids in
eccentric annuli has been investigated numerically. Effects of eccentricity on frictional
pressure loss in annulus for different fluids are presented. Numerical results are compared
with previous studies. Numerical investigation based on SIMPLE algorithm by Patankar-
Spalding (1972)

has been presented for the case of Newtonian fluid flow in eccentric
annulus with inner pipe rotation.
Yield power-law rheology model is used as the constitutive equation of the flow.
Cartesian and boundary fitted coordinate system are utilized as two different approaches
to discretize the flow equations and generate mesh network. A minimum cut off value for
iv
the minimum shear rate is used to identify the plug (zero shear rate) region. Fluid flow
equations have been solved using an iterative successive over relaxation method.
Increasing eccentricity is found to lower frictional pressure drop for different
fluids. It was observed that for non-Newtonian fluids the effect of eccentricity on
pressure loss is less pronounced compared to Newtonian fluids.

v

ACKNOWLEDGEMENTS

The author thanks Dr. Mengjiao Yu, dissertation advisor, for his continuous
patience and assistance in this endeavor.

My special thanks to Dr. Ramadan Ahmed for his help and valuable suggestions.

I am thankful to Dr. Siamack Shirazi and Dr. Stefan Miska for their help,
suggestions and encouragement.

My appreciation extends to National Iranian Oil Company for providing me with
financial support.

I also thank TUDRP students and all my friends for their constant encouragement
and support.

3.1.7 Grid refinement in tangential direction (number of grids in radial direction=60).
Comparison of the value for volume flow rate of simulation with analytical solutions for
concentric annulus for a Newtonian (n=1) and a power-law (n=0.2) fluid. ........................30

3.1.8 Grid refinement in radial direction (number of grids in tangential direction=60).
Comparison of the value for volume flow rate of simulation with analytical solutions for
concentric annulus for a Newtonian (n=1) and a power-law (n=0.2) fluid. ........................31

4.1.4 Eccentricity vs. error in pressure drop of the different studies compared with Piercy et al.
for a Newtonian fluid....................................................................................................50

1.1 Background
During drilling operation of oil wells a fluid is pumped from a surface mud tank
down to the bottom of the well through drill pipe, through nozzles in the drill bit and then
back to the surface mud tank trough the annular space between the drill pipe and well
bore wall as shown in Fig.1.1. The drilling fluid has to satisfy several requirements such
as: supporting the well bore wall from collapsing, preventing formation fluid from
transferring to the well bore, cooling the drill bit, carrying cutting from bottom of the
well to the surface through annulus, etc.
The pump pressure is a function of pressure losses in surface equipment, in drill
pipe, across the bit nozzles, through the annulus, etc. Frictional pressure loss through
annulus is the main concern of this study.
Several factors can affect frictional pressure loss in annulus such as flow rate,
flow regime, mud rheology, well bore geometry, cuttings content, drill pipe rotation, drill
pipe lateral motion or swirling introduced by the rotation itself and/or fluid flow, etc. In
this study eccentricity is the main investigated issue along with rheology and drill pipe
rotation.
2

1.2 Significance of the Subject
In conventional drilling, frictional pressure loss accounts for about 10% of the
whole circulation pressure loss.
[15]

In slim hole configurations, annular frictional pressure loss can contribute 30-50%
of the total fluid circulation pressure loss and some investigators have reported it to be as
high as 90%.
[8]
Significant frictional pressure drop in the annulus can result in an increase of the
bottom hole pressure, which might then exceed the strength of the formation, thus
causing loss of drilling fluid and creating a potentially dangerous situation due to the
resulting loss of hydrostatic head.
[15]

Fig.1.1 Schematic of Drilling Operation
3
1.3 Contribution and Evaluation of This Study
In this study effect of eccentricity of drill pipe and rheology of drilling fluid on
frictional pressure drop in annuli for Newtonian and non-Newtonian fluid is investigated.
A computer program in FORTRAN is developed to perform calculations.
Results of the program are compared with the present works of Azouz
[3]
, Escudier
[4, 5]
,
Piercy
[17]
and Haciislamoglu
[7]
and found to be all in good agreement. Also, numerical
simulation for a Newtonian annular flow in eccentric annulus with a rotational drill pipe
is performed.

1.4 Scope of the Study
A short review of the past works related to this study is the content of Chapter 2.
This review has categorized the previous studies according to the type of the problem.
Chapter 3 is an investigation of laminar flow of Newtonian and non-Newtonian
fluids in eccentric annulus. In the first part of this chapter, drill pipe is considered to be
stationary and fluid is Newtonian or non-Newtonian. The second part of the chapter
considers the drill pipe to be rotating and fluid to be Newtonian only. In the first part of
the chapter, two approaches are used to solve the problem. For the first approach a
rectangular mesh system is used to solve discretized flow equations in Cartesian system.
In the second approach, on the other hand, a boundary fitted coordinate system is utilized
to apply numerical technique to equations. Flow equations as well as geometry are
transformed into this new system and then discretized. Similar solution procedures are
used to solve the equations in Cartesian and boundary fitted coordinate system. The last
part of Chapter 3 is related to flow of Newtonian fluid in eccentric annulus with a rotating
drill pipe. Navier-Stokes equations of motion and continuity in a rectangular coordinate
4
system are transformed into a computational plane in a boundary fitted coordinate
system. SIMPLE algorithm by Patankar-Spalding
[16]
is chosen to solve the equations.
In chapter 4 results of numerical simulation are presented and compared with
other studies. Results are related to Newtonian and non-Newtonian fluid flow in eccentric
annulus with stationary drill pipe. Results for the case when drill pipe is in motion are not
presented due to instability of the numerical procedure.

5

CHAPTER 2

LITERATURE REVIEW

This chapter is a review of some the previous work about laminar annular flow.
Laminar annular flow is flow of a fluid in an annulus under laminar regime. In laminar
regime, fluid particles travel along well-ordered non-intersecting paths, or layers.
The determination of annular flow performance is important in the planning and
design of the hydraulic program for a well not only because of significant pressure losses
but also possible hole cleaning problems.
Laminar annular flow has been the subject of many investigations. With the
developments in computers and also computational fluid dynamics there is a high trend
of using numerical simulations.

2.1 Annular Flow of Newtonian Fluid
One of the first correlations for annular flow is given by Lamb (1932)
[12]
. In that
study, an equation is presented that correlates frictional pressure loss with flow rate in a
concentric annulus for a Newtonian fluid.
( )
1
1
1
]
1

Where,
o
R and
i
R are the outer and inner cylinder radii and c is the displacement of the
two centers. Momentum equation in the rectangular Cartesian coordinates for this case to
be solved is:
Fig.2.1.1 Newtonian fluid in concentric annulus
R
i

¸
¸
+ − − +
·
· −
−
−
∆
∆
i
o
i o
a b
R R
L
P
y
bR
aR
R R a b
ab
a b
i o
ln
2
2 2
2 2
2 2
τ
6 . 2 . 2 . eq
Hanks (1979)
[9]
presented analytical solution for yield power- law fluid. In that
study several charts are presented through computing theoretical solutions of the
equations of the motion for the concentric annular geometry using the Herschel-Bulkly
model. Using those charts to find some parameters and applying them to the equations,
one can compute volume flow rate through given pressure drop and also computing
pressure drop through given volume flow rate.

2.3 Eccentric No-Rotation Non-Newtonian
Haciislamoglu (1985)
[7]
and Azouz (1994)
[3]
used numerical simulation to solve
the governing equations of non-Newtonian fluid flow in eccentric annulus without pipe
rotation.
Nearly all of the analytical and numerical solutions for annular flow without pipe
rotation indicate that frictional pressure loss decreases with increasing eccentricity, at a
12
given flow rate. The reduction in pressure loss can be significant for instant can be in the
range of the pressure loss.

2.4 Newtonian Fluid Flow in Annulus with Inner Pipe Rotation
Yamada (1960)
[22]
described the experimental results for the resistance of water
flow through an annulus formed by two concentric cylinders, with the inner cylinder
rotating and the outer cylinder stationary. This study shows that when the flow is laminar,
the resistance of a flow is unaffected up to a certain rotational speed. But beyond this
speed the flow resistance increases as the Reynolds number increases.
Ooms et al. (1996)
[15]
performed numerical simulation of Newtonian fluids flow
in annulus to investigate effects of eccentricity and rotational speed of the inner pipe on
frictional pressure loss. Their study was more accurately followed by Escudier et al.
(1999)
[4]
. Theoretical and experimental study of Escudier et al.
[4]
suggested that in the
case of concentric annulus, rotation of the inner pipe does not influence frictional
pressure loss of the Newtonian fluids. However, in the case of eccentric annulus they
showed that when the inner pipe is rotating, for low eccentricities (less than 30%),
pressure drop remains approximately constant with increasing of eccentricity. For higher
eccentricities however, frictional pressure loss generally decreases with increasing of
eccentricity. Escudier et al.
[4]
also found that for a given radius ratio, as rotation speed of
the inner cylinder increases at any eccentricity, frictional pressure loss increases.

13
2.5 Concentric With Rotation Non-Newtonian
Pilehvari (1989)
[18]
studied laminar, helical flow of a power law fluid in a
concentric annulus with inner pipe rotation. He adopted the differential equations
developed by Direkes and Schowlter and solved them by using the finite element method.
The results of their study shows that drill pipe rotation, lowers the pressure drop at
constant flow rate. He also pointed out that this effect is likely to be offset by formation
of Taylor vortices which form at higher pipe rotations.

to propose a new model of power law
fluids in eccentric annuli. They showed that drill pipe rotation can reduce annular
frictional pressure loss due to the shear-thinning effect. He mentioned that overall annular
frictional pressure loss is the combined results of shear thinning effects and pipe lateral
motion or axial vibration effect and in most cased, the latter is the dominant factor.
Hussain and Sharif (2000)
[11]
performed a numerical investigation of Herschel-
Bulkley fluids in concentric and eccentric annuli with rotating inner cylinder. In their
study, they used a finite volume algorithm with a nonstaggered grid system and a non-
orthogonal curvilinear coordinate system to handle irregular geometry of an eccentric
annulus to analyze the problem. They found an increase of the flow rate with increasing
eccentricity for a fixed axial pressure gradient and pipe rotation speed.
Escudier et al. (2001)
[5]
performed numerical calculations using a finite volume
method for the fully developed laminar flow of a shear-thinning power- law fluid through
an eccentric annulus with inner cylinder rotation. It is shown that in general, pressure
14
drop values for power- law fluids follow the trends observed by Escudier et al. (1999)
[4]

for Newtonian fluids, including an increase with rotation of the inner pipe, an increase
with eccentricity at low and very high eccentricities but a decrease for intermediate
eccentricities. They showed that a power- law fluid generally exhibits lower pressure drop
as compared with the Newtonian liquid.

15

CHAPTER 3

ANNULAR FLOW IN ECCENTRIC ANNULUS

In this chapter laminar flow of Newtonian and non- Newtonian fluids through an
eccentric annulus is numerically simulated. In the first part of the chapter drill pipe is
considered to be stationary and fluid is Newtonian or non-Newtonian. In the second part
of this chapter inner pipe is rotating and fluid is Newtonian.

3.1 Annular Flow in Eccentric Annulus without Inner Pipe Rotating

To simulate fluid flow in an eccentric annulus when the drill pipe is stationary,
governing equations for flow in eccentric annulus are developed in the Cartesian
coordinate system. Yield Power Law model is used to represent the rheology of the fluid.
To solve the flow equations, two approaches are used. In the first approach a rectangular
mesh system is used to solve discretized flow equations in the Cartesian coordinate
system. In the second approach, on the other hand, a boundary fitted coordinate system is
utilized to apply numerical technique to equations.
In the first approach using a rectangular grid system, equations are discretized and
an iterative over relaxation method is used to solve for velocity field at each grid point
knowing frictional pressure loss in axial direction of the well bore. A relaxation technique
is used to speed up convergence of the solution. Calculation of relaxation factor is
presented and the treatment for the no shear region is discussed.
16
In the second approach similar solution procedure is used, but geometry as well as
equations are transformed into a computational domain in a boundary fitted coordinate
system.

3.1.1 Assumptions
1-Laminar and fully developed flow
2-Isothermal and steady state conditions
3-Flowing direction is in the annulus along the axial direction of the well bore.
4- Incompressible fluid
5- Flow domain is an eccentric annulus.
6- Drill pipe is stationary
7-No slippage at the walls

¸
¸
∂
∂
∂
∂
µ µ 3 . 1 . 3 . eq
Where, w is the fluid velocity in z direction and µ is the viscosity that is a
function of shear rate. The viscosity function depends on the rheology of the fluid. The
shear rate for axial flow can be expressed as:
2 2

Yield Power Law (Herschel-Bulkley) Model:
Pipe and annular flows of Yield Power-Law fluid is of great interests in drilling
applications. This model describes the rheological behavior of drilling muds more
accurately than Bingham Plastic and Power-Law models. The Yield Power-Law
rheological model for all time- independent fluids is given by:
n
y
Kγ τ τ & + ·

A non-Newtonian fluid does not have a constant viscosity like Newtonian. However, in
numerical modeling, the concept of viscosity is used for non-Newtonian fluids to make
the governing equations similar to Newtonian fluids. This viscosity is known as apparent
20
viscosity that is a function of shear rate and depends on the rheology of fluid. Apparent
viscosity is generally defined as:

3.1.5 Numerical Procedure
To solve the equation of motion ( 12 . 1 . 3 . eq ), a numerical method based on finite
difference technique is used. In order to apply this technique, the flow region is
subdivided into a grid network. Fig 3.1.3 shows the grid network in a Cartesian
coordinate system centered on the center of the drill pipe.
Since the grid points do not coincide exactly with the circular boundaries, some
interpolations are made near the boundaries. The more grid points we use, the more
accurate geometry will be achieved. Since the geometry is symmetrical around y axis, we
only need to obtain the numerical solution for half of the domain.
21

Fig.3.1.3 Grid network in Cartesian coordinate with different grid sizes in x and y direction

3.1.6 Discretizing the Equation of Motion

Equation of motion is a differential equation that can be approximated by a
discretized finite difference equation. A second-order central differencing scheme has
been used for all the grid points inside of the flow domain. This scheme establishes the
following system of algebraic equations:

Where:
j i
n
W
,
1 +
: The value of the velocity field at node j i, at computational step (iteration) 1 + n
j i
n
W
,
: The value of the velocity field at node j i, at computational step (iteration) n
Where, ω is over relaxation factor and is calculated following Azouz
[3]
at each
grid point and it is shown in (Appendix D):

3.1.8 Minimum Shear Rate
Visco-plastic fluids encounter a problem for small shear rate in the un-yielded
region while calculating the apparent viscosity. This is the most difficult aspect for
numerical modeling of fluids with yield stress. Azouz
[3]
used a minimum cut off value of
the shear rate. It is better to determine a cut off value corresponding to the conditions. It
24
was experienced that if the cut of value is too small or too big, sometimes we need too
many unnecessary iterations or we may encounter instability problem. The following
equations can be used for calculating cut off value of minimum shear rate:
¹
¹
¹
¹
¹
¹
¹
¹
¹
'
¹

,
_

¸
¸
·
−
Stress Yiel without Fluids for
Stress Yield with Fluids for
K
wall
n
y
γ
τ
γ
&
&
8
1
8
min
10
10
18 . 1 . 3 . eq
where,
wall
γ& can be calculated from 24 . .B eq .Detailed derivations of the above equations
can be found in Appendix B.
Iteration begins with a relatively large value of cut off shear rate. When velocity
field reaches a convergence the cut off value must decrease and iterations continue with
the new cut off shear rate. This procedure continues until the smallest desired cut off
value is obtained. From then on, it will remain constant.

3.1.9 Convergence Criteria
When the values of velocity field do not change with more iteration and also
smallest calculated shear rate reaches the minimum desired shear rate, convergence is
achieved and calculations should be stopped.
Convergence of velocity is achieved when velocity field at new iterations is close
to the values of the previous iteration. This is defined by a relative value, called residual
of velocity:
25
( ) ( )
( )
∑∑
∑∑
· ·
· ·
−
·
max max
max max
1 1
,
1 1
, ,
j
j
i
i
j i
old
j
j
i
i
j i
old
j i
new
w
w w
Residual Velocity 19 . 1 . 3 . eq
If the velocity residual is small enough( )
7
10 . .
−
g e , convergence is achieved.

One advantage the boundary fitted coordinate system compared to rectangular
coordinate system is its ability to conform to the boundaries of the system regardless of
i
i+1 i-1
j+1
j
j-1
y ∆

x ∆

j i
w
,

y x A
j i
∆ ∆ ·
,
26
the shape. In another word grid generated can fit itself into the boundary of the system as
shown in Fig.3.1.4. This feature increases the accuracy of the solution developed.
Doing the transformation, physical Cartesian coordinates (x,y) becomes the
dependent variables and the curvilinear coordinates ( η ξ, ) becomes the independent
variables. A generated grid is then defined as a set of points formed by the intersections
of the lines of a boundary conforming curvilinear coordinate system.

Fig.3.1.5 Grid Network in Boundary Fitted Coordinate System

There is a uniform grid in this system in the sense that at each radius position the
angular width is constant and at each angular position, the radial width of the cells is also
constant.

3.1.15 Grid Refinement Analysis
A gird refinement analysis showed that using grid numbers more than 100 in
radial and in tangential direction does not result in a significant change in flow rate. In
order to check the results for different grid numbers, a concentric annulus was considered
and the results of the numerical simulation was compared with the results of analytical
solution for Newtonian and non-Newtonian as it is shown in 6 . 1 . 3 . Fig and 7 . 1 . 3 . Fig .

Fig. 3.1.7 Grid refinement in tangentialdirection (number of grids in radial direction=60).
Comparison of the value for volume flow rate of simulation with analytical solutions for
concentric annulus for a Newtonian (n=1) and a power-law (n=0.2) fluid.

Fig. 3.1.8 Grid refinement in radial direction (number of grids in tangential direction=60).
Comparison of the value for volume flow rate of simulation with analytical solutions for
concentric annulus for a Newtonian (n=1) and a power-law (n=0.2) fluid.

The objective of this section is to investigate laminar Newtonian fluid flow in a
concentric annulus with inner pipe rotation. Navier-Stokes equations for laminar flow of
a Newtonian fluid in Cartesian coordinate system are considered. All the equations are
transformed to a computational domain in the boundary fitted coordinate system.
Geometry, on the other hand is transformed algebraically. Having equations and
33
geometry in the computational domain, SIMPLE algorithm (Patankar –Spalding, 1972) is
applied to solve the equations.

3.2.1 Assumptions

1-Laminar and fully developed flow
2-Isothermal and steady state conditions
3-Flowing direction is in the annulus along the axial direction of the well bore.
4- Incompressible fluid
5- Flow domain is an eccentric annulus.
6- Drill pipe is rotating at a constant angular velocity
7-No slippage at the walls

¸
¸
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
z
w
y
v
x
u
t
ρ
3 . 2 . 3 . eq
34
The first term on the left hand side of all the momentum equations is zero because
the flow is under steady state. Also on the right hand side of the same equations the last
term in the second parenthesis is zero due to flow is fully developed. Also the last term
on 1 . 2 . 3 . eq and 2 . 2 . 3 . eq is zero because there is not any gravity in x and y directions.
In continuity equation, the first term is also zero because of a steady state flow.
The last term on the left hand side of this equation is also zero because fluid is fully
developed.
Gravity in
y x,
directions is zero. Also we wish to eliminate gravity in
z
direction
in order to consider only frictional pressure loss so
4 . 2 . 3 21 . 3 . − eq
will take the new form:
( ) ( )
x
p
y
v
x
u
z
w u
y
v u
x
u
∂
∂
−
1
1
]
1

This set of equations ( ) 2 . 22 . 3 19 . 2 . 3 . − eqs exhibit a mixed elliptic-parabolic
behavior, and hence the standard relaxation technique which is widely used for those
types of equations is not particularly helpful.
To solve these kinds of equations, Patankar and Spalding (1972) proposed a
method called SIMPLE (Semi Implicit Pressure Linked Equation) algorithm. SIMPLE
algorithm is basically an iterative approach, where some innovative physical reasoning is
used to make the next iteration from the results of the previous iteration. The idea is to
start with discrete continuity equation and substitute into it the discrete u and v
momentum equations. Discrete momentum equations contain pressure differences hence
we can get an equation for the discrete pressures. SIMPLE algorithm actually solves for a
related quantity called the pressure correction.

3.2.6 Discretization of Momentum and Continuity Equation

Using ordinary discretization of equations in these types of equations may create a
checkerboarding problem as described by Patankar-Spalding
[15]
. A popular remedy for
checkerboarding is the use of a staggered mesh. The key feature here is to calculate
pressure and velocity at different grid points as shown in 2 . 2 . 3 . Fig .

39

Fig.3.2.2 Staggered grid for velocity and pressure

The original formulation of the SIMPLE method by Patankar and Spalding
involved a finite-volume approach. In this study, a finite-difference approach is used
which would essentially give the same results as obtained by finite-volume method. We
choose to use a forward or backward differencing for the grid points close to the walls.
For other points, central differencing with second order accuracy is used. The
computational domain is assumed to have equal spaces in η ξ , direction meaning:
1 · ∆ · ∆ η ξ .

The primary idea behind SIMPLE is to create a discrete equation for pressure (or
alternatively, a related quantity called the pressure correction) from the discrete
continuity equation( ) 27 . 2 . 3 . eq . Since the continuity equation contains discrete face
velocities, we need some way to relate these discrete velocities to the discrete pressure
field. The SIMPLE algorithm uses the discrete momentum equations to derive this
connection. Let
* *
,V U and
*
W be the discreteU ,V andW fields resulting from a solution of
the discrete V U, andW momentum equations. Let
*
P represent the discrete pressure field
which is used in the solution of the momentum equations. Thus
1 , , 1
,
− − j i j i
V U and
j i
W
,
satisfy:

4.1.1 Results for Newtonian Fluid
As eccentricity increases, frictional pressure drop decreases. This value could be
about 50% less than the pressure drop of a concentric annulus with the same flow rate.
When the annulus becomes eccentric, one side of annuls is wider than the other side.
Since fluid always tends to bulge through the wider area, most of the fluid flows through
the wider area of the annulus (as shown in Fig.4.1.1), where there is less restriction. In
other words the portion of the fluid that is taking the highest restriction of the geometry is
less than the portion under lower restriction. Figure.4.1.3 shows the results of the
simulation for Newtonian fluid in eccentric annuli.

Figure 4.1.4 shows the comparison of the present and past studies with the
analytical solution of Piercy et al.
[17]
. All the cases can predict quite reasonable results.
Considering the trend of the error for different studies, in the study of Azouz
[3]
error
seems to be quite constant for different eccentricities. However this is quite different for
the other two studies where with increasing of the eccentricity, error decreases. Since the
error can be generate from different sources, one can not explain the reason of these
discrepancies.
50
Present Study
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.2 0.4 0.6 0.8 1 1.2
Eccentricity
E
r
r
o
r
%
Ri/Ro=0.8
Ri/Ro=0.5
Ri/Ro=0.2
Escudier et al.
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0 0.2 0.4 0.6 0.8 1 1.2
Eccentricity
E
r
r
o
r
%
Ri/Ro=0.8
Ri/Ro=0.5
Ri/Ro=0.2
Azouz

Fig.4.1.4 Eccentricity vs. error in pressure drop of the different studies compared with
Piercy et al. for a Newtonian fluid

51
4.2 Power-Law Fluid

4.2.1 Results for Power-Law Fluid
As it is shown in figures 4.2.1-4.2.4, the same trend that was observed for
Newtonian fluids can be found for the power-law fluid as well. It seems that for power-
law fluids the effect of eccentricity on pressure loss is less pronounced compared to
Newtonian fluids. As fluids become more shear thinning (decreasing n), their velocity
profiles in the wide and narrow parts of the annulus becomes flatter; thus increasing their
overall viscosity. Consequently, these fluids are subject to less reduction in frictional
pressure losses in eccentric annulus.

Like power- law and Newtonian fluids, the same trend is observed for yield
power- law fluid. For the following case (Fig. 4.3.1), reduction in frictional pressure loss
for eccentric annulus compared to concentric case is 40%. A comparison between the
results of this study with Haciislamoglu
[7]
for this case shows a good agreement between
the two studies and the differences is less than 2%.

Following is a comparison between the experimental results of Ahmed
[1]
for
bentonite in a fully eccentric annulus. The difference between the results was
considerably high and up about 15%. This discrepancy can be due to the time
dependency of some of bentonite or the reason could be transition to turbulent flow.
58
0
10
20
30
40
50
60
70
0 5 10 15 20 25
Q[GPM]
P
r
e
s
s
u
r
e

Note: Extensive numerical simulation using the program indicates that the
numerical procedure used in this study is not able to accurately predict the flow rate for
fluids with yield stress when the plug region is relatively large.

4.5 Conclusions

Ø For a given flow rate increasing eccentricity of the inner pipe decreases the
pressure drop in annuli;
59
Ø Pressure loss reduction due to eccentricity for highly shear thinning ( ) 2 . 0 · n
non-Newtonian fluids is approximately 30%, while for Newtonian fluids can
be as high as 50%;
Ø The reduction of pressure loss due to eccentricity, is more significant at higher
radius ratios

,
_

¸
¸
→1 . .
o
i
R
R
e i ;
Ø Using Cartesian grid networks simplifies the equation of motion and
numerical procedure, however at the same time less accuracy was observed;
Ø Due to complexity of the pipe rotation case, it is recommended to do more
simplification to find the source of instability.

Residual in 15 . .B eq will decrease as we approach the solution. And when the
convergence is achieved, value of residual will reach very close to zero. Based on this fact we can
increase/decrease rate of convergence by multiplying residual by an over/under relaxation factor
shown by ω . As a result, 14 . . B eq will take the form:
70
( )

¸
¸
·
τ
γ& 22 . .B eq
Equation 22 . B can not be used for fluids without yield stress, because the value will be
zero. So a new cut off value has to be defined. For fluids with zero yield stress, wall shear rate
can be used as a value to estimate the cut off value. As a reasonable approximation:
wall
γ γ & &
8
min
10
−
· 23 . .B eq
71
Where,
wall
γ& is in the order of average wall shear rate for annular flow and can be
calculated as:
h
wall
D
V 8
· γ& 24 . .B eq
Where, V is mean velocity in annulus and
h
D is hydraulic diameter for a concentric
annulus and can be expressed as:
( )
i o h
R R D − · 2
25 . .B eq

Where,
o
R is well bore radius and
i
R is outer radius of drill pipe. To calculateV , narrow
slot approximation solution can be used. Referring to Miska
[14]
for fluids with constant density
the relationship between shear stress and pressure gradient
l
p
∂
∂
in a rectangular slot is:

,
_

¸
¸
∂
∂
− ·
l
p
y τ
26 . .B eq Where, y is a vertical distance from the center of the rectangular slot. If h is the
height of the slot, at the wall
2
h
y · and then:

Considering Fig.F.1:
4
,
3
,
2
,
1
,
,
4 4 4 4
A
W
A
W
A
W
A
W
Q
j i j i j i j i
j i
+ + + · 1 . .F eq
Where
j i
Q
,
is the small flow rate element at one grid point,
j i
W
,
is the velocity
field at one grid point and
1
A -
4
A are shown in Fig.F.2.
Each area is made by four points, so each point has only
4
1
of an area therefore,
( )
4 3 2 1
,
,
4
A A A A
W
Q
j i
j i
+ + + · 2 . .F eq